Auger bits need to be run at lowish speed in 25mm size, and that is only a 12 volt too, so not a lot of torque, Larger auger bits, over 12mm normally have a 7/16in (11.1mm) hex shank. I just confirmed this by measuring a few of my own. So anything with a 10mm chuck won't be big enough for an auger bit (just as well given the high power/torque requirements of such drills). TBH even if you could get a smaller shank auger bit I doubt that would have the beans to drill through a 2in joist with a 25mm auger bit very often without the risk of either knackering the gearbox or cooking the motor. I use one of these for cabinet installation (Milwaukee M12FPDXKIT), it has a 13mm chuck and whilst it can d 25mm holes in 2in joists it labours and gets pretty hot - and it's a brushless tool:
It will, however, drive a 32mm spade bit through a 38mm laminated chipboard worktop, so I agree with
@cdbe and
@foxhole that the OP would be far better off switching to a spade bit rather than an auger bit and running at higher speed - it doesn't require such a powerful motor, just a fast one, and you don't need a tool with a 13mm chuck. Just keep the tool sharp if you are drilling many (needle file). And, OP, I know that what I showed was an angle impact drill, but with 1/4in hex shank spade bits it will indeed drill 2in softwood joists with smaller bits up to about 20mm (never tried bigger ones, but TBH you didn't specify your requirement for 25mm holes through joists). Despite being pretty gutless it has the speed, and I do drill smaller holes with it periodically when I don't have a heavier angle drill with me (fortunately my spade bits are all 1/4in hex shank), but it really isn't my tool of choice for joist drilling
You can get all sorts of drill bits in 1/4in hex shank these days:
But always bear in mind that auger bits require a lot more power/torque than spade bits. You can even get 1/4in hex drive auger bits up to 25mm or so, the problem is that they have a tendency to wind up and snap like carrots in the larger sizes, so I no longer use them (sorry, Trend).
Take it from me, NO drill adaptor, other than the heavy duty ones I first linked to (mounted on a reasonably powerful drill) will ever be able to run a 25mm auger bit through joists reliably and consistently without breaking. Despite what you may think, this really IS a heavy-duty application. FYI for heavy applications the trade has started moving on to tools like this (Makita DDA460) which will run 30mm auger bits or 50mm hole hog bits through joists with ease:

,
But they are expensive, and they weigh a ton, especially when used above your shoulder level.
Bosch, Fischer, etc